Logics

Traditional degree semantics approaches have aimed to pin down the inherent class of adjectives. This paper presents a novel dynamic perspective, where the classification of an adjective is dynamic and syntactically dependent. Using measurement theory and fuzzy set analysis, the proposed framework defines dynamic patterns of adjective classes with a set of axioms and integrates these patterns with syntactic structures to explain the flexibility and constraints observed in adjectival expressions. Employing Mandarin data, the paper illustrates how different syntactic constructions select specific adjective classes, thereby affecting their distribution and interpretation. This approach not only accommodates cross-linguistic variations but also provides a more comprehensive understanding of the semantics of adjectives. Full article

We offer a framework that captures both context-dependency and vagueness of predicate meanings—illustrated by the politically relevant case of woman—as an interaction of lexical meaning and Question under Discussion (‘QUD’). We extend existing approaches to non-maximality to superficially polysemous predicates like woman and show that this is conceptually coherent and insightful for a linguistic analysis of political debates about gender invitation policies: while there are (i) clear, semantically true, and (ii) strictly unacceptable cases of x is a woman, there are also (iii) merely pragmatically acceptable cases (‘like a woman with respect to the QUD’) as well as (iv) truly vague ones. We argue that this four-way division is the least complex model that captures current gender discourses in a harm-reducing, trans-inclusive way. This offers a new perspective on the semantics–pragmatics interface of predicate meanings. Full article

José Morgado introduced in 1962 a novel notion of hyperlattices, which he called reticuloides. In his master’s thesis submitted in 1971 (under the supervision of Newton da Costa), Antonio M. Sette introduced a new class of implicative hyperlattices (here called SIHLs) based on Morgado’s hyperlattices. He also extended SIHLs by adding a unary hyperoperator, thus defining a class of hyperalgebras (denoted ${\mathrm{SHC}}_{\omega }$) corresponding to da Costa algebras for $C_{\omega }$, thereby providing suitable semantics for the logic $C_{\omega }$. In this paper, after providing a (hyper)lattice-theoretic characterization of Sette’s implicative hyperlattices and proving some basic results on SIHLs, we introduce a class of swap structures—special hyperalgebras over the signature of $C_{\omega }$ that arise naturally from implicative lattices. We prove that these swap structures are indeed ${\mathrm{SHC}}_{\omega }$. Finally, we demonstrate that the class ${\mathrm{SHC}}_{\omega }$, as well as the aforementioned swap structures, characterizes the logic $C_{\omega }$. Full article

If one believes that $2+2=4$, then one also believes that either $2+2=4$ or 971 is a cousin prime number. This follows from doxastic logics based on standard Kripke relational semantics, which validate disjunction introduction for belief. However, this principle does not hold in topic-sensitive semantics. An agent who lacks the concept of a ‘cousin prime number’ may be unable to entertain, and thus unable to believe, any proposition involving that concept. I argue that while disjunction introduction may fail for belief—and for other epistemic states that presuppose belief—it does hold for certain states that do not require belief. In this paper, I focus on the notion of commitment to the truth. Drawing on the concept of logical grounding, I propose formal semantics that preserve the requirement of topic-grasping, but weaken it in a way that allows for a more standard treatment of disjunction. Full article

Rationality has long been considered the quintessence of humankind. However, psychological experiments revealing reliable divergences in performances on reasoning tasks from normative principles of reasoning have cast serious doubt on the venerable dogma that human beings are rational animals. According to the standard picture, reasoning in accordance with principles based on rules of logic, probability theory, etc., is rational. The standard picture provides the backdrop for both the rationality and irrationality thesis, and, by virtue of the competence-performance distinction, diametrically opposed interpretations of reasoning experiments are possible. However, the standard picture rests on shaky foundations. Jean Piaget developed a psychological theory of reasoning, in which logic and mathematics are continuous with psychology but nevertheless autonomous sources of knowledge. Accordingly, logic, probability theory, etc., are not extra-human norms, and reasoners have the ability to reason in accordance with them. In this paper, I set out Piaget’s theory of rationality, using intra- and interpropositional reasoning as illustrations, and argue that Piaget’s theory of rationality is compatible with the standard picture but actually undermines it by denying that norms of reasoning based on logic are psychologically relevant for rationality. In particular, rather than logic being the normative benchmark, I argue that rationality according to Piaget has a psychological foundation, namely the reversibility of the operations of thought constituting cognitive structures. Full article

Analogies are an important part of human cognition for learning and discovering new concepts. There are many different approaches to defining analogies and how new ones can be found or constructed. We propose a novel approach in the tradition of structure mapping using colored multigraphs to represent domains. We define a category of colored multigraphs in order to utilize some Category Theory (CT) concepts. CT is a powerful tool for describing and working with structure-preserving maps. There are many useful applications for this theory in cognitive science, and we want to introduce one such application to a broader audience. CT and the concepts used in this paper are introduced and explained. We show how the category theoretical concepts product and pullback can be used with the category of colored multigraphs to find possible analogies between domains using different requirements. The dual notion of a pullback, the pushout, is then used as conceptual blending to generate a new domain. Full article

Inquisitive modal logic ${\mathrm{InqML}}_{⊞}$ is a natural generalization of basic modal logic, with ⊞ as a primitive modal operator. In this paper, we study the bisimulation quotients in the logic ${\mathrm{InqML}}_{⊞}$. For a given inquisitive modal model $\mathfrak{M}=(W,\Sigma ,V)$, we first show that the bisimilarity relation is an equivalence relation on W and that there is the largest bisimulation on $\mathfrak{M}$. We then define the bisimulation quotient and prove that a model is connected to its bisimulation quotient by a surjective bounded morphism. Finally, we prove that two models are globally bisimilar if and only if their bisimulation quotients are isomorphic. Full article

We present an extension and generalization of Sahlqvist–van Benthem correspondence to the case of distribution-free modal logic, with, or without negation and/or implication connectives. We follow a reductionist strategy, reducing the correspondence problem at hand to the same problem, but for a suitable system of sorted modal logic (the modal companion of the distribution-free system). The reduction, via a fully abstract translation, builds on the duality between normal lattice expansions and sorted residuated frames with relations (a generalization of classical Kripke frames with relations). The approach is scalable and it can be generalized to other systems, with or without distribution, such as distributive modal logic, or substructural logics with, or without additional modal operators. Full article

This work contextualizes the possibility of deriving a unifying artificial intelligence framework by walking in the footsteps of General, Explainable, and Verified Artificial Intelligence (GEVAI): by considering explainability not only at the level of the results produced by a specification but also considering the explicability of the inference process as well as the one related to the data processing step, we can not only ensure human explainability of the process leading to the ultimate results but also mitigate and minimize machine faults leading to incorrect results. This, on the other hand, requires the adoption of automated verification processes beyond system fine-tuning, which are essentially relevant in a more interconnected world. The challenges related to full automation of a data processing pipeline, mostly requiring human-in-the-loop approaches, forces us to tackle the framework from a different perspective: while proposing a preliminary implementation of GEVAI mainly used as an AI test-bed having different state-of-the-art AI algorithms interconnected, we propose two other data processing pipelines, LaSSI and EMeriTAte+DF, being a specific instantiation of GEVAI for solving specific problems (Natural Language Processing, and Multivariate Time Series Classifications). Preliminary results from our ongoing work strengthen the position of the proposed framework by showcasing it as a viable path to improve current state-of-the-art AI algorithms. Full article

Logics of Statements in Context have been proposed as a general framework to describe and relate, in a uniform and unifying way, a broad spectrum of logics and specification formalisms, which also comprise “open formulas”. In particular, it has been shown that we can define arbitrary first-order “open formulas” in arbitrary categories. At present, there are two deficiencies. In the general case, only signatures with predicate symbols are considered and institutions of statements in context are only defined for single signatures. In this paper, we elaborate the special case of traditional many-sorted first-order logic. We show that any many-sorted first-order signature $\mathrm{\Sigma }$ with predicates and (!) operation symbols gives rise to an institution ${\mathcal{FL}}_{\mathrm{\Sigma }}$ of $\mathrm{\Sigma }$-statements in context and that any signature morphism results in a comorphism between the corresponding institutions. We prove that we obtain a functor $\mathcal{FL}:\mathsf{Sig}\to co\mathbb{I}ns$ from the category of signatures into the category of institutions and comorphisms. We construct a corresponding Grothendieck institution ${\mathcal{FL}}^{♯}$ and prove that ${\mathcal{FL}}^{♯}$ is, indeed, an extension of the traditional institution of first-order logic, which only comprises “closed formulas”. We also investigate substitutions in detail and discuss (elementary) diagrams in the sense of traditional first-order logic. Full article

I aim to examine Val Plumwood’s feminist account of logic, as presented by Plumwood, using the frameworks developed by Elliott and McKaughan, and Intemann. Plumwood argues that relevance logic is the appropriate logical system for feminist reasoning. I intend to assess whether this constitutes a legitimate incorporation of values into logic. To this end, I evaluate the aims of Plumwood as a case study. Additionally, I trace the values embedded in my chosen case to determine whether feminist values advance the epistemic and social objectives of the research. Full article

We study knowable informational dependence between empirical questions, modeled as continuous functional dependence between variables in a topological setting. We also investigate epistemic independence in topological terms and show that it is compatible with functional (but non-continuous) dependence. We then proceed to study a stronger notion of knowability based on uniformly continuous dependence. On the technical logical side, we determine the complete logics of languages that combine general functional dependence, continuous dependence, and uniformly continuous dependence. Full article

We present Stratified Metric Temporal Logic (SMTL), a novel formalism for specifying and verifying the properties of complex cyber–physical systems that exhibit behaviors across multiple temporal and abstraction scales. SMTL extends existing temporal logics by incorporating a stratification operator, enabling the association of temporal properties with specific abstraction levels. This allows for the natural expression of multi-scale requirements while maintaining formal reasoning about inter-level relationships. We formalize the syntax and semantics of SMTL, proving that it strictly subsumes metric temporal logic (MTL) and offers enhanced expressiveness by capturing properties unattainable in existing logics. Numerical simulations comparing agents operating under MTL and SMTL specifications show that SMTL enhances agent coordination and safety, reducing collision rates without substantial computational overhead or compromising path efficiency. These findings highlight SMTL’s potential as a valuable tool for designing and verifying complex multi-agent systems operating across diverse temporal and abstraction scales. Full article

In this paper, we study three algorithmic problems involving computation trees: the optimization, solvability, and satisfiability problems. The solvability problem is concerned with recognizing computation trees that solve problems. The satisfiability problem is concerned with recognizing sentences that are true in at least one structure from a given set of structures. We study how the decidability of the optimization problem depends on the decidability of the solvability and satisfiability problems. We also consider various examples with both decidable and undecidable solvability and satisfiability problems. Full article

This article initiates the semantic study of distribution-free normal modal logic systems, laying the semantic foundations and anticipating further research in the area. The article explores roughly the same area, though taking a different approach, as a recent article by Bezhanishvili, de Groot, Dmitrieva and Morachini, who studied a distribution-free version of Dunn’s positive modal logic (PML). Unlike PML, we consider logics that may drop distribution and that are equipped with both an implication connective and modal operators. We adopt a uniform relational semantics approach, relying on recent results on representation and duality for normal lattice expansions. We prove canonicity and completeness in the relational semantics of the minimal distribution-free normal modal logic, assuming just the K-axiom, as well as those of its axiomatic extensions obtained by adding any of the D, T, B, S4 or S5 axioms. Adding distribution can be easily accommodated and, as a side result, we also obtain a new semantic treatment of intuitionistic modal logic. Full article

This article contributes to recent results in the model theory of distribution-free logics (which include a Goldblatt-Thomason theorem and a development of their Sahlqvist theory) by lifting van Benthem’s characterization result for modal logic to the more general setting of the logics of normal lattice expansions. Our proof approach makes use of a fully abstract translation of the language of the logics of interest into the language of sorted residuated modal logic, building on an analogous translation of substructural logics recently published by the author. The article is intended as a demonstration and application of a project of reduction of non-distributive logics to (sorted) residuated modal logics. The reduction makes the proof of a van Benthem characterization of non-distributive logics possible, by adapting, reusing and generalizing existing results. Full article

Homophily, which means similarity breeds association, is one of the most fundamental principles in social organization. However, in some cases, homophily is not significant, because actors’ perceptions of others differ from the real situation. In this article, we use the term “subjective homophily” to describe the homophily where the perceived similarity of objects is considered. In addition, we also consider social influence, which is closely related to homophily and represents the diffusion of some attributes through associations. In short, the dynamic temporal logic ${\mathrm{LoSH}}_{\mathrm{G},\mathrm{M}}^{\mathrm{SC}}$ we propose in this article is based on computation tree logic (CTL), which is used to describe the evolution of networks by subjective homophily, and dynamic logic (DL), which provides the dynamic update operator for representing active social influence. Furthermore, we prove that the model checking problem and the validity checking problem for ${\mathrm{LoSH}}_{\mathrm{G},\mathrm{M}}^{\mathrm{SC}}$ are both PSPACE-complete. Finally, we use an example, named false consensus, to illustrate how logic captures the subjective homophily evolution of networks and the impact of active social influence on evolution and structure. Full article